Optical laws are undoubtedly a wonderful principle, and in nature, there exists another "axiom" that we can see everywhere. In our daily lives, we always see the phenomenon of "water flowing to lower ground." This is the result of water being in the Earth’s gravitational field (and it is precisely because of this that the suicide activities of certain despondent individuals can be carried out smoothly; of course, we do not need to personally experiment to verify this point.). From this, we can think of a term: gravitational potential energy. What does "water flowing to lower ground" mean? The height becomes lower. What does a lower height mean? The gravitational potential energy has decreased! In other words, objects in nature have a tendency to move toward the lowest potential energy. We can explain it from this perspective: Systems always tend toward stability, and the higher the energy (potential energy) they possess, the more unstable they are.
Speaking of which, we have obtained another extremum with obvious practical significance. However, we cannot yet discover exactly how to turn it into something calculable. Indeed, we must also cooperate with an "Equilibrium Axiom" to build the "bridge connecting physics and mathematics":
When the total potential energy of the system reaches its minimum, there must be a force equilibrium (net external force is 0).
In fact, this is very clear. If the net external force were not zero, there would certainly be a force driving the system to move in a direction of lower energy. However, the converse of this axiom is not necessarily true. Of course, this does not affect our application of it.
Systems always tend toward stability, and the higher the energy (potential energy) they possess, the more unstable they are.
Supplement:
Regarding the Equilibrium Axiom, it seems we can no longer discuss much more content, because it is truly content we are all too familiar with. So in this section, let us briefly summarize Fermat’s Principle and the Equilibrium Axiom: both are reflections of common phenomena in life. Starting from these two principles to answer certain problems, we are often captivated by their concise beauty. Some might think this is a bit like "juggling in the Olympiads," but that is not the case. Physics can guide us to think correctly, while mathematics helps us summarize and analyze conclusions. Mathematics is very scientific, but what is truly magical is physics. Physical science has shocked humanity time and again. God is an artist, and the world He created is so harmonious. As the saying goes: Chemistry is physics without thought. Mathematics is physics without purpose.
In the subsequent research on problems such as the catenary and the brachistochrone (later we will find that the essence of these types of problems is the same), Euler and Lagrange developed a highly dynamic branch of mathematics called "calculus of variations." This was both unexpected and exactly what everyone had hoped for.